* the last coefficient can be Order(_ex1()) to represent a truncated,
* non-terminating series.
*
- * @param var_ series variable (must hold a symbol)
- * @param point_ expansion point
+ * @param rel__ expansion variable and point (must hold a relational)
* @param ops_ vector of {coefficient, power} pairs (coefficient must not be zero)
* @return newly constructed pseries */
-pseries::pseries(const ex &var_, const ex &point_, const epvector &ops_)
- : basic(TINFO_pseries), seq(ops_), var(var_), point(point_)
+pseries::pseries(const ex &rel_, const epvector &ops_)
+ : basic(TINFO_pseries), seq(ops_)
{
- debugmsg("pseries constructor from ex,ex,epvector", LOGLEVEL_CONSTRUCT);
- GINAC_ASSERT(is_ex_exactly_of_type(var_, symbol));
+ debugmsg("pseries constructor from rel,epvector", LOGLEVEL_CONSTRUCT);
+ GINAC_ASSERT(is_ex_exactly_of_type(rel_, relational));
+ GINAC_ASSERT(is_ex_exactly_of_type(rel_.lhs(),symbol));
+ point = rel_.rhs();
+ var = *static_cast<symbol *>(rel_.lhs().bp);
}
n.add_ex("point", point);
}
-
-/*
- * Functions overriding virtual functions from base classes
- */
+//////////
+// functions overriding virtual functions from bases classes
+//////////
basic *pseries::duplicate() const
{
void pseries::print(ostream &os, unsigned upper_precedence) const
{
debugmsg("pseries print", LOGLEVEL_PRINT);
- convert_to_poly().print(os, upper_precedence);
+ for (epvector::const_iterator i=seq.begin(); i!=seq.end(); i++) {
+ // omit zero terms
+ if (i->rest.is_zero())
+ continue;
+ // print a sign, if needed
+ if (i!=seq.begin())
+ os << '+';
+ if (!is_order_function(i->rest)) {
+ // print 'rest', i.e. the expansion coefficient
+ if (i->rest.info(info_flags::numeric) &&
+ i->rest.info(info_flags::positive)) {
+ os << i->rest;
+ } else
+ os << "(" << i->rest << ')';
+ // print 'coeff', something like (x-1)^42
+ if (!i->coeff.is_zero()) {
+ os << '*';
+ if (!point.is_zero())
+ os << '(' << var-point << ')';
+ else
+ os << var;
+ if (i->coeff.compare(_ex1())) {
+ os << '^';
+ if (i->coeff.info(info_flags::negative))
+ os << '(' << i->coeff << ')';
+ else
+ os << i->coeff;
+ }
+ }
+ } else {
+ os << Order(power(var-point,i->coeff));
+ }
+ }
}
void pseries::printraw(ostream &os) const
{
- debugmsg("pseries printraw", LOGLEVEL_PRINT);
- os << "pseries(" << var << ";" << point << ";";
- for (epvector::const_iterator i=seq.begin(); i!=seq.end(); i++) {
- os << "(" << (*i).rest << "," << (*i).coeff << "),";
- }
- os << ")";
+ debugmsg("pseries printraw", LOGLEVEL_PRINT);
+ os << "pseries(" << var << ";" << point << ";";
+ for (epvector::const_iterator i=seq.begin(); i!=seq.end(); i++) {
+ os << "(" << (*i).rest << "," << (*i).coeff << "),";
+ }
+ os << ")";
+}
+
+void pseries::printtree(ostream & os, unsigned indent) const
+{
+ debugmsg("pseries printtree",LOGLEVEL_PRINT);
+ os << string(indent,' ') << "pseries "
+ << ", hash=" << hashvalue << " (0x" << hex << hashvalue << dec << ")"
+ << ", flags=" << flags << endl;
+ for (unsigned i=0; i<seq.size(); ++i) {
+ seq[i].rest.printtree(os,indent+delta_indent);
+ seq[i].coeff.printtree(os,indent+delta_indent);
+ if (i!=seq.size()-1) {
+ os << string(indent+delta_indent,' ') << "-----" << endl;
+ }
+ }
+ var.printtree(os, indent+delta_indent);
+ point.printtree(os, indent+delta_indent);
}
unsigned pseries::nops(void) const
ex pseries::coeff(const symbol &s, int n) const
{
if (var.is_equal(s)) {
- if (seq.size() == 0)
- return _ex0();
-
- // Binary search in sequence for given power
- numeric looking_for = numeric(n);
- int lo = 0, hi = seq.size() - 1;
- while (lo <= hi) {
- int mid = (lo + hi) / 2;
- GINAC_ASSERT(is_ex_exactly_of_type(seq[mid].coeff, numeric));
- int cmp = ex_to_numeric(seq[mid].coeff).compare(looking_for);
- switch (cmp) {
- case -1:
- lo = mid + 1;
- break;
- case 0:
- return seq[mid].rest;
- case 1:
- hi = mid - 1;
- break;
- default:
- throw(std::logic_error("pseries::coeff: compare() didn't return -1, 0 or 1"));
- }
- }
- return _ex0();
+ if (seq.size() == 0)
+ return _ex0();
+
+ // Binary search in sequence for given power
+ numeric looking_for = numeric(n);
+ int lo = 0, hi = seq.size() - 1;
+ while (lo <= hi) {
+ int mid = (lo + hi) / 2;
+ GINAC_ASSERT(is_ex_exactly_of_type(seq[mid].coeff, numeric));
+ int cmp = ex_to_numeric(seq[mid].coeff).compare(looking_for);
+ switch (cmp) {
+ case -1:
+ lo = mid + 1;
+ break;
+ case 0:
+ return seq[mid].rest;
+ case 1:
+ hi = mid - 1;
+ break;
+ default:
+ throw(std::logic_error("pseries::coeff: compare() didn't return -1, 0 or 1"));
+ }
+ }
+ return _ex0();
} else
return convert_to_poly().coeff(s, n);
}
ex pseries::collect(const symbol &s) const
{
- if (var.is_equal(s))
- return convert_to_poly();
- else
- return inherited::collect(s);
+ return *this;
}
+/** Evaluate coefficients. */
ex pseries::eval(int level) const
{
if (level == 1)
return this->hold();
+ if (level == -max_recursion_level)
+ throw (std::runtime_error("pseries::eval(): recursion limit exceeded"));
+
// Construct a new series with evaluated coefficients
epvector new_seq;
new_seq.reserve(seq.size());
new_seq.push_back(expair(it->rest.eval(level-1), it->coeff));
it++;
}
- return (new pseries(var, point, new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
+ return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
}
-/** Evaluate numerically. The order term is dropped. */
+/** Evaluate coefficients numerically. */
ex pseries::evalf(int level) const
{
- return convert_to_poly().evalf(level);
+ if (level == 1)
+ return *this;
+
+ if (level == -max_recursion_level)
+ throw (std::runtime_error("pseries::evalf(): recursion limit exceeded"));
+
+ // Construct a new series with evaluated coefficients
+ epvector new_seq;
+ new_seq.reserve(seq.size());
+ epvector::const_iterator it = seq.begin(), itend = seq.end();
+ while (it != itend) {
+ new_seq.push_back(expair(it->rest.evalf(level-1), it->coeff));
+ it++;
+ }
+ return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
}
ex pseries::subs(const lst & ls, const lst & lr) const
{
- // If expansion variable is being substituted, convert the series to a
- // polynomial and do the substitution there because the result might
- // no longer be a power series
- if (ls.has(var))
- return convert_to_poly(true).subs(ls, lr);
-
- // Otherwise construct a new series with substituted coefficients and
- // expansion point
- epvector new_seq;
- new_seq.reserve(seq.size());
- epvector::const_iterator it = seq.begin(), itend = seq.end();
- while (it != itend) {
- new_seq.push_back(expair(it->rest.subs(ls, lr), it->coeff));
- it++;
- }
- return (new pseries(var, point.subs(ls, lr), new_seq))->setflag(status_flags::dynallocated);
+ // If expansion variable is being substituted, convert the series to a
+ // polynomial and do the substitution there because the result might
+ // no longer be a power series
+ if (ls.has(var))
+ return convert_to_poly(true).subs(ls, lr);
+
+ // Otherwise construct a new series with substituted coefficients and
+ // expansion point
+ epvector new_seq;
+ new_seq.reserve(seq.size());
+ epvector::const_iterator it = seq.begin(), itend = seq.end();
+ while (it != itend) {
+ new_seq.push_back(expair(it->rest.subs(ls, lr), it->coeff));
+ it++;
+ }
+ return (new pseries(relational(var,point.subs(ls, lr)), new_seq))->setflag(status_flags::dynallocated);
}
/** Implementation of ex::diff() for a power series. It treats the series as a
}
it++;
}
- return pseries(var, point, new_seq);
+ return pseries(relational(var,point), new_seq);
} else {
return *this;
}
/** Default implementation of ex::series(). This performs Taylor expansion.
* @see ex::series */
-ex basic::series(const symbol & s, const ex & point, int order) const
+ex basic::series(const relational & r, int order) const
{
epvector seq;
numeric fac(1);
ex deriv = *this;
- ex coeff = deriv.subs(s == point);
+ ex coeff = deriv.subs(r);
+ const symbol *s = static_cast<symbol *>(r.lhs().bp);
+
if (!coeff.is_zero())
seq.push_back(expair(coeff, numeric(0)));
int n;
- for (n=1; n<order; n++) {
+ for (n=1; n<order; ++n) {
fac = fac.mul(numeric(n));
- deriv = deriv.diff(s).expand();
+ deriv = deriv.diff(*s).expand();
if (deriv.is_zero()) {
// Series terminates
- return pseries(s, point, seq);
+ return pseries(r, seq);
}
- coeff = fac.inverse() * deriv.subs(s == point);
+ coeff = fac.inverse() * deriv.subs(r);
if (!coeff.is_zero())
seq.push_back(expair(coeff, numeric(n)));
}
// Higher-order terms, if present
- deriv = deriv.diff(s);
+ deriv = deriv.diff(*s);
if (!deriv.is_zero())
seq.push_back(expair(Order(_ex1()), numeric(n)));
- return pseries(s, point, seq);
+ return pseries(r, seq);
}
/** Implementation of ex::series() for symbols.
* @see ex::series */
-ex symbol::series(const symbol & s, const ex & point, int order) const
+ex symbol::series(const relational & r, int order) const
{
- epvector seq;
- if (is_equal(s)) {
- if (order > 0 && !point.is_zero())
- seq.push_back(expair(point, _ex0()));
- if (order > 1)
- seq.push_back(expair(_ex1(), _ex1()));
- else
- seq.push_back(expair(Order(_ex1()), numeric(order)));
- } else
- seq.push_back(expair(*this, _ex0()));
- return pseries(s, point, seq);
+ epvector seq;
+ const ex point = r.rhs();
+ GINAC_ASSERT(is_ex_exactly_of_type(r.lhs(),symbol));
+ const symbol *s = static_cast<symbol *>(r.lhs().bp);
+
+ if (this->is_equal(*s)) {
+ if (order > 0 && !point.is_zero())
+ seq.push_back(expair(point, _ex0()));
+ if (order > 1)
+ seq.push_back(expair(_ex1(), _ex1()));
+ else
+ seq.push_back(expair(Order(_ex1()), numeric(order)));
+ } else
+ seq.push_back(expair(*this, _ex0()));
+ return pseries(r, seq);
}
if (!is_compatible_to(other)) {
epvector nul;
nul.push_back(expair(Order(_ex1()), _ex0()));
- return pseries(var, point, nul);
+ return pseries(relational(var,point), nul);
}
// Series addition
}
}
}
- return pseries(var, point, new_seq);
+ return pseries(relational(var,point), new_seq);
}
/** Implementation of ex::series() for sums. This performs series addition when
* adding pseries objects.
* @see ex::series */
-ex add::series(const symbol & s, const ex & point, int order) const
+ex add::series(const relational & r, int order) const
{
ex acc; // Series accumulator
// Get first term from overall_coeff
- acc = overall_coeff.series(s, point, order);
-
+ acc = overall_coeff.series(r, order);
+
// Add remaining terms
epvector::const_iterator it = seq.begin();
epvector::const_iterator itend = seq.end();
if (is_ex_exactly_of_type(it->rest, pseries))
op = it->rest;
else
- op = it->rest.series(s, point, order);
+ op = it->rest.series(r, order);
if (!it->coeff.is_equal(_ex1()))
op = ex_to_pseries(op).mul_const(ex_to_numeric(it->coeff));
new_seq.push_back(*it);
it++;
}
- return pseries(var, point, new_seq);
+ return pseries(relational(var,point), new_seq);
}
if (!is_compatible_to(other)) {
epvector nul;
nul.push_back(expair(Order(_ex1()), _ex0()));
- return pseries(var, point, nul);
+ return pseries(relational(var,point), nul);
}
-
+
// Series multiplication
epvector new_seq;
ex a_coeff = coeff(*s, i);
ex b_coeff = other.coeff(*s, cdeg-i);
if (!is_order_function(a_coeff) && !is_order_function(b_coeff))
- co += coeff(*s, i) * other.coeff(*s, cdeg-i);
+ co += a_coeff * b_coeff;
}
if (!co.is_zero())
new_seq.push_back(expair(co, numeric(cdeg)));
}
if (higher_order_c < INT_MAX)
new_seq.push_back(expair(Order(_ex1()), numeric(higher_order_c)));
- return pseries(var, point, new_seq);
+ return pseries(relational(var,point), new_seq);
}
/** Implementation of ex::series() for product. This performs series
* multiplication when multiplying series.
* @see ex::series */
-ex mul::series(const symbol & s, const ex & point, int order) const
+ex mul::series(const relational & r, int order) const
{
ex acc; // Series accumulator
// Get first term from overall_coeff
- acc = overall_coeff.series(s, point, order);
+ acc = overall_coeff.series(r, order);
// Multiply with remaining terms
epvector::const_iterator it = seq.begin();
acc = ex_to_pseries(acc).mul_const(ex_to_numeric(f));
continue;
} else if (!is_ex_exactly_of_type(op, pseries))
- op = op.series(s, point, order);
+ op = op.series(r, order);
if (!it->coeff.is_equal(_ex1()))
op = ex_to_pseries(op).power_const(ex_to_numeric(it->coeff), order);
}
if (!higher_order && !all_sums_zero)
new_seq.push_back(expair(Order(_ex1()), numeric(deg) + p * ldeg));
- return pseries(var, point, new_seq);
+ return pseries(relational(var,point), new_seq);
}
/** Implementation of ex::series() for powers. This performs Laurent expansion
* of reciprocals of series at singularities.
* @see ex::series */
-ex power::series(const symbol & s, const ex & point, int order) const
+ex power::series(const relational & r, int order) const
{
ex e;
if (!is_ex_exactly_of_type(basis, pseries)) {
// Basis is not a series, may there be a singulary?
if (!exponent.info(info_flags::negint))
- return basic::series(s, point, order);
+ return basic::series(r, order);
// Expression is of type something^(-int), check for singularity
- if (!basis.subs(s == point).is_zero())
- return basic::series(s, point, order);
+ if (!basis.subs(r).is_zero())
+ return basic::series(r, order);
// Singularity encountered, expand basis into series
- e = basis.series(s, point, order);
+ e = basis.series(r, order);
} else {
// Basis is a series
e = basis;
/** Re-expansion of a pseries object. */
-ex pseries::series(const symbol & s, const ex & p, int order) const
-{
- if (var.is_equal(s) && point.is_equal(p)) {
- if (order > degree(s))
- return *this;
- else {
- epvector new_seq;
- epvector::const_iterator it = seq.begin(), itend = seq.end();
- while (it != itend) {
- int o = ex_to_numeric(it->coeff).to_int();
- if (o >= order) {
- new_seq.push_back(expair(Order(_ex1()), o));
- break;
- }
- new_seq.push_back(*it);
- it++;
- }
- return pseries(var, point, new_seq);
- }
- } else
- return convert_to_poly().series(s, p, order);
+ex pseries::series(const relational & r, int order) const
+{
+ const ex p = r.rhs();
+ GINAC_ASSERT(is_ex_exactly_of_type(r.lhs(),symbol));
+ const symbol *s = static_cast<symbol *>(r.lhs().bp);
+
+ if (var.is_equal(*s) && point.is_equal(p)) {
+ if (order > degree(*s))
+ return *this;
+ else {
+ epvector new_seq;
+ epvector::const_iterator it = seq.begin(), itend = seq.end();
+ while (it != itend) {
+ int o = ex_to_numeric(it->coeff).to_int();
+ if (o >= order) {
+ new_seq.push_back(expair(Order(_ex1()), o));
+ break;
+ }
+ new_seq.push_back(*it);
+ it++;
+ }
+ return pseries(r, new_seq);
+ }
+ } else
+ return convert_to_poly().series(r, order);
}
/** Compute the truncated series expansion of an expression.
- * This function returns an expression containing an object of class pseries to
- * represent the series. If the series does not terminate within the given
+ * This function returns an expression containing an object of class pseries
+ * to represent the series. If the series does not terminate within the given
* truncation order, the last term of the series will be an order term.
*
- * @param s expansion variable
- * @param point expansion point
+ * @param r expansion relation, lhs holds variable and rhs holds point
* @param order truncation order of series calculations
* @return an expression holding a pseries object */
-ex ex::series(const symbol &s, const ex &point, int order) const
+ex ex::series(const ex & r, int order) const
{
GINAC_ASSERT(bp!=0);
- return bp->series(s, point, order);
+ ex e;
+ relational rel_;
+
+ if (is_ex_exactly_of_type(r,relational))
+ rel_ = ex_to_relational(r);
+ else if (is_ex_exactly_of_type(r,symbol))
+ rel_ = relational(r,_ex0());
+ else
+ throw (std::logic_error("ex::series(): expansion point has unknown type"));
+
+ try {
+ e = bp->series(rel_, order);
+ } catch (exception &x) {
+ throw (std::logic_error(string("unable to compute series (") + x.what() + ")"));
+ }
+ return e;
}
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